Dilip V. Sarwate. A note on universal classes of hash functions. Inf. Process. Lett., 10(1):41--45, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....only if the authentication matrix is an orthogonal array OA( k; where = k( 1) 1) and the authentication rules are used with equal probability. 3 Universal hashing Universal classes of hash functions were introduced by Carter and Wegman [2] and were studied further by Sarwate [6], Wegman and Carter [14] and Stinson [9] In this paper, we are interested in the application of universal hashing to authentication codes. First, let us review the relevant de nitions. Let A and B be nite sets, and denote a = jAj and b = jBj, where a b. A function h : A B will be termed a ....

....1=b; and given that x 1 is mapped to y 1 , the conditional probability that x 2 is mapped to y 2 is at most , for any x 2 ; y 2 ; x 2 6= x 1 . Special cases of these de nitions have been previously studied in the literature. For example, a b) ab b) AU has been called optimally universal or OU [6], 1=b) AU has been called universal or U (cf. Theorem 3.1) 2] and (1=b) ASU has been called strongly universal or SU [14] Note that in an ASU class, it must be the case that 1=b. Also, if = 1=b, then property 2(b) implies property 2(a) ASU classes of hash functions can be used for ....

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D. V. Sarwate. A note on universal classes of hash functions. Information Processing Letters 10 (1980), 41-45.

....by Carter and Wegman [5] ffl The case ffl = n Gamma m) m(n Gamma 1) is known as optimally universal hashing. The reason for this terminology is that ffl (n Gamma m) m(n Gamma 1) in any ffl U (N ; n; m) hash family; see Corollary 3.7. This definition was first given in 1980 by Sarwate [21]. Here are some more definitions. DEFINITION Suppose that the functions in an (N ; n; m) hash family, F , have range Y = G, where G is an additive abelian group (of order m) F is called ffl Delta universal provided that for any two distinct elements x 1 ; x 2 2 X and for any element y 2 G, ....

....families. Not surprisingly, we will employ well known bounds from coding theory. The Plotkin bound (see, for example [17, p. 58] implies that D N K(q Gamma 1) K Gamma 1)q in any (N; K;D; q) code. It can be used to derive the following lower bound on ffl that was first proved by Sarwate [21]. Theorem 3.3 If there exists an ffl U (N ; n; m) hash family, then ffl n Gamma m m(n Gamma 1) Proof. Using Theorem 3.1, construct an (N; n; N(1 Gamma ffl) m) code from the hash family. This code must satisfy the Plotkin bound, so we obtain N(1 Gamma ffl) N n(m Gamma 1) n Gamma ....

D. V. Sarwate, A Note on Universal Classes of Hash Functions. Information Processing Letters 10 (1980), 41--45.

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Dilip V. Sarwate. A note on universal classes of hash functions. Inf. Process. Lett., 10(1):41--45, 1980.

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D. V. Sarwate. A note on universal classes of hash functions, Inform. Proc. Letters 10 (1980), 41-45.

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